Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes

Abstract

We establish estimates for the asymptotic best approximation of the Euclidean unit ball by polytopes under a notion of distance induced by the intrinsic volumes. We also introduce a notion of distance between convex bodies that is induced by the Wills functional, and apply it to derive asymptotically sharp bounds for approximating the ball in high dimensions. Remarkably, it turns out that there is a polytope which is almost optimal with respect to all intrinsic volumes simultaneously, up to absolute constants. Finally, we establish asymptotic formulas for the best approximation of smooth convex bodies by polytopes with respect to a distance induced by dual volumes, which originate from Lutwak's dual Brunn-Minkowski theory.

arXiv comment44 pages, 1 figure. To appear in International Mathematics Research Notices

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Cited by in corpus (3)

• An intrinsic volume metric for the class of convex bodies in ${\mathbb{R}}^n$
• Surface areas of equifacetal polytopes inscribed in the unit sphere ${\mathbb{S}}^2$
• Perimeter approximation of convex discs in the hyperbolic plane and on the sphere